Linear combinations of the individual atomic orbitals (LCAO)
Linear combinations of the individual atomic orbitals (LCAO):
- In Fig. given below we schematically show the coulombic potential wells of two atoms close to each other, along with the wave functions of two electrons centered on the two nuclei.
- By solving the Schrbdinger equation for such an interacting system, we find that the composite two-electron wave functions are linear combinations of the individual atomic orbitals (LCAO).
- The odd or antisymmetric combination is called the antibonding orbital, while the even or symmetric combination is the bonding orbital.
- It can be seen that the bonding orbital has a higher value of the wave function (and therefore the electron probability density) than the antibonding state in the region between the two nuclei.
- This corresponds to the covalent bond between the atoms.
- To determine the energy levels of the bonding and the antibonding states, it is important to recognize that in the region between the two nuclei the coulombic potential energy V(r) is lowered (solid line in Fig.) compared to isolated atoms (dashed lines).
- It is easy to see why the potential energy would be lowered in this region, because an electron here would be attracted by two nuclei, rather than just one.
- For the bonding state the electron probability density is higher in this region of lowered potential energy than for the antibonding state.
- As a result, the original isolated atomic energy level would be split into two, a lower bonding energy level and a higher antibonding level.
- It is the lowering of the energy of the bonding state that gives rise to cohesion of the crystal.
- For even smaller interatomic spacings, the energy of the crystal goes up because of repulsion between the nuclei, and other electronic interactions.
- Since the probability density is given by the square of the wave function, if the entire wave function is multiplied by -1, it does not lead to a different LCAO.
- The important point to note in this discussion is that the number of distinct LCAO, and the number of distinct energy levels, depends on the number of atoms that are brought together.
The lowest energy level corresponds to the totally symmetric LCAO, the highest corresponds to the totally antisymmetric case, and the other combinations lead to energy levels in between.